Hearing damage may occur when a person is exposed to a sound intensity level of 90.0 dB (relative to the threshold of hearing) for a period of 9.0 hours. One particular eardrum has an area of 1.90 10-4 m2. How much sound energy is incident on this eardrum during this time?
Sound intensity level in dB assigned to sound intensity J is:
L = 10 ∙ log₁₀(J/J₀)
where J₀ = 10⁻¹²W/m² is th threshold of hearing
Hence:
J = J₀ ∙ 10^(L/10)
So the intensity of a 90dB sound is:
J = 10⁻¹²W/m² ∙ 10^(90/10) = 10⁻³W/m²
Multiply by the eardrum area and you get the power input to it.
P = J ∙ A
Multiply this by the time of exposure (in seconds) and you have the incident energy:
E = P ∙ t = J ∙ A ∙ t
= 10⁻³ ∙ 1.9 ∙10⁻⁴ ∙ 9 ∙3600 =
= 6.15•10⁻³J
To calculate the sound energy incident on the eardrum, we can use the formula:
Sound Energy = Sound Intensity × Time
First, let's convert the sound intensity level from dB to its corresponding intensity in watts per square meter (W/m^2). The formula to convert from dB to intensity is:
I = (10^(L/10)) × I₀
Where I is the intensity in W/m^2, L is the sound intensity level in dB, and I₀ is the reference intensity, which is 1.0 × 10^(-12) W/m^2.
Given that the sound intensity level is 90.0 dB, we can substitute the values into the formula:
I = (10^(90.0/10)) × (1.0 × 10^(-12))
Next, let's calculate the sound intensity (I):
I = (10^(9.0)) × (1.0 × 10^(-12))
I = 1.0 × 10^(-3) W/m^2
Now that we have the sound intensity, we can calculate the sound energy incident on the eardrum using the formula:
Sound Energy = Sound Intensity × Area × Time
Given that the area of the eardrum is 1.90 × 10^(-4) m^2 and the time of exposure is 9.0 hours, we can substitute the values into the formula:
Sound Energy = (1.0 × 10^(-3) W/m^2) × (1.90 × 10^(-4) m^2) × (9.0 hours)
To convert the time from hours to seconds, we multiply by 3600 seconds:
Sound Energy = (1.0 × 10^(-3) W/m^2) × (1.90 × 10^(-4) m^2) × (9.0 hours) × (3600 seconds/hour)
Sound Energy = (1.0 × 10^(-3) W/m^2) × (1.90 × 10^(-4) m^2) × (9.0) × (3600)
Finally, we can calculate the sound energy incident on the eardrum:
Sound Energy ≈ 6.12 joules
To calculate the sound energy incident on the eardrum, we need to use the formula:
Sound Energy = Sound Intensity × Area × Time
First, let's convert the threshold of hearing from dB to W/m^2 using the equation:
I = 10^(dB/10) × I0
Where I0 represents the threshold of hearing, which is typically 1.0 × 10^(-12) W/m^2.
Using this equation, we can find the sound intensity I:
I = 10^(90/10) × 1.0 × 10^(-12) W/m^2
Next, we can calculate the sound energy by multiplying the sound intensity by the area and time:
Sound Energy = I × Area × Time
Now, let's plug in the given values:
Sound Energy = (10^(90/10) × 1.0 × 10^(-12) W/m^2) × (1.90 × 10^(-4) m^2) × (9.0 hours)
Before we proceed, note that 1 hour is equivalent to 3600 seconds. Therefore, we need to convert the time from hours to seconds:
Time (in seconds) = 9.0 hours × 3600 seconds/hour
Now we can calculate the sound energy:
Sound Energy = (10^(90/10) × 1.0 × 10^(-12) W/m^2) × (1.90 × 10^(-4) m^2) × (9.0 hours × 3600 seconds/hour)
Next, let's perform the calculations step by step:
Threshold of hearing (I0) = 1.0 × 10^(-12) W/m^2
Sound intensity (I) = 10^(90/10) × (1.0 × 10^(-12)) W/m^2
Time (in seconds) = 9.0 hours × 3600 seconds/hour
Sound Energy = (10^(90/10) × (1.0 × 10^(-12)) W/m^2) × (1.90 × 10^(-4) m^2) × (9.0 hours × 3600 seconds/hour)
Calculating the values within the parentheses:
Threshold of hearing (I0) = 1.0 × 10^(-12) W/m^2
Sound intensity (I) = 10^(90/10) × 1.0 × 10^(-12) W/m^2
≈ 1.0 × 10^(-3) W/m^2
Time (in seconds) = 9.0 hours × 3600 seconds/hour
= 3.24 × 10^4 seconds
Substituting the values:
Sound Energy = (1.0 × 10^(-3) W/m^2) × (1.90 × 10^(-4) m^2) × (3.24 × 10^4 seconds)
Performing the multiplication:
Sound Energy ≈ 1.0 × 10^(-3) × 1.90 × 10^(-4) × 3.24 × 10^4 Joules
Finally, we can calculate the product of these values:
Sound Energy ≈ 6.21 × 10^(-3) Joules
Therefore, approximately 6.21 × 10^(-3) Joules of sound energy is incident on the eardrum during this time.